Finite Group Databasebeta

← #341669

Subgroups — #341669

Order 512.

498 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28799C212yesmaximal
2#26442C212yesmaximal
3#10686C211yesmaximal, characteristic
4#10656C212yesmaximal
5#10574C212yesmaximal
6#10352C211yesmaximal, characteristic
7#9575C212yesmaximal
8#9538C212yesmaximal
9#8929C211yesmaximal, characteristic
10#2571C2211yescharacteristic
11#2551C2212yes—
12#2369C2211yescharacteristic
13#2088—22no—
14#2085—22no—
15#2058—22no—
16#2058—22no—
17#1993—22no—
18#1990C2212yes—
19#1988—22no—
20#1987C2212yes—
21#1956C2211yescharacteristic
22#1948C2212yes—
23#1948C2212yes—
24#1947C2212yes—
25#1943—22no—
26#1939C2212yes—
27#1919—22no—
28#1844C2211yescharacteristic
29#1834C2212yes—
30C4×#106C2212yes—
31#1770C2212yes—
32C2×#333C2211yescharacteristic
33#1296C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2311yescharacteristic
42C2×#107—22no—
43C2×#106—22no—
44C2×#106—22no—
45C2×#106—41no—
46C2×#106D412yes—
47C2×#106D412yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—41no—
52C2×#97—22no—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2312yes—
55#400—41no—
56#395—42no—
57#391—41no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#384C2311yescharacteristic
63#384C2312yes—
64#382—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2312yes—
67C4×C4⋊C4C2312yes—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#333Q814yescomplemented
82C22×Q8—21no—
83C22×Q8—42no—
84C22×C8—22no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89C4⋊Q8—22no—
90C4⋊Q8—21no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93C4⋊Q8—82no—
94C4⋊Q8—82no—
95C4⋊Q8C2×Q812yes—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—41no—
113C2×C42C2411yescharacteristic
114#107—44no—
115#106—24no—
116#106—42no—
117#106—82no—
118#106—82no—
119#106—82no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—82no—
124C4⋊C8C2×Q814yes—
125#103—82no—
126#97—82no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130#95—22no—
131#95—22no—
132#95—42no—
133#95—41no—
134#95—42no—
135#95—42no—
136C2×Q8—22no—
137C2×Q8—41no—
138C2×Q8—84no—
139C2×Q8—81no—
140C2×Q8—81no—
141C2×Q8—84no—
142C2×Q8—162no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4C22×D412yes—
151C22×C4H(5,𝔽2)11yescharacteristic
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—24no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C4⋊C4—22no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—84no—
183C4⋊C4—81no—
184C4⋊C4—82no—
185C4⋊C4—84no—
186C4⋊C4—161no—
187C4⋊C4—161no—
188C4⋊C4#13714yes—
189C42—42no—
190C42—42no—
191C42—42no—
192C42—42no—
193C42—161no—
194C42C22×Q811yescharacteristic
195C422−1+411yescharacteristic
196C422−1+412yes—
197C23D4211yescharacteristic
198Q8—82no—
199Q8—161no—
200Q8—168no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4#53612yes—
224C2×C4#54411yescharacteristic
225C2×C4D4×Q812yes—
226C2×C4#56311yescharacteristic
227C2×C4C2×#13712yes—
228C2×C4#57512yes—
229C2×C4#57712yes—
230C2×C4#57712yes—
231C8—42no—
232C8—44no—
233C8—42no—
234C8—42no—
235C8—82no—
236C22#267711yescharacteristic
237C22#326612yes—
238C22#327512yes—
239C22#328811yescharacteristic
240C22#329111yescharacteristic
241C4—21no—
242C4—21no—
243C4—44no—
244C4—82no—
245C4—82no—
246C4—82no—
247C4—82no—
248C4—161no—
249C4—162no—
250C4#323712yes—
251C4#324412yes—
252C4#338712yes—
253C4#338812yes—
254C2#2010711yescharacteristic
255C2#2017712yes—
256C2#2029411yescharacteristic
257C2#3096912yes—
258C2#3097911yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
81. #333Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#287992
#264422
#106862
#106562
#105742
#103522
#95752
#95382
#89292